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Lin I.H. — Geometric Linear Algebra. Vol. 1

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Название: Geometric Linear Algebra. Vol. 1

Автор: Lin I.H.

Аннотация:

Writing for freshmen and sophomores in college or advanced high school students, Lin (National Taiwan Normal U.) takes the view that recently geometric linear algebra has become too algebraic and tackles the subject from another angle, starting with the affine and linear structures of real vector spaces R1, R2 and R3, stepping through them as one-dimensional (including affine variants), two- dimensional (including linear transformations and affine transformations) and three dimensions, including lines and planes in space. In appendices Lin gives the background information necessary to follow the text and do the exercises, including an extended description of algebraic linear algebra.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 856

Добавлена в каталог: 04.07.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Linear transformation (mapping) right invertible (inverse)      141
Linear transformation (mapping), linear functional      84 345 420 732
Linear transformation (mapping), linear operator      84 732
Linear transformation (mapping), linear operator, restriction operator      89 210
Linear transformation (mapping), linear operator, restriction operator, direct sum      211
LS-decomposition      167
LU-Decomposition      161 167 442
Markov (or stochastic) process (or chain)      487 501
Markov (or stochastic) process (or chain), limiting probability vector      503
Markov (or stochastic) process (or chain), nonnegative matrix      502
Markov (or stochastic) process (or chain), positive matrix      501
Markov (or stochastic) process (or chain), positive stochastic matrix      501 504
Markov (or stochastic) process (or chain), probability vector      487 501
Markov (or stochastic) process (or chain), regular stochastic matrix      487 501
Markov (or stochastic) process (or chain), stability or steady-state vector      503
Markov (or stochastic) process (or chain), transition (or stochastic or Markov) matrix      501
Matrix      37 46 47 54 73 86 336 342 343 364 366 399 699
Matrix acting as a linear transformation      86 737
Matrix entry (element)      699
Matrix ith row      699
Matrix jth column      700
Matrix norm (of a matrix)      490 498 546 557
Matrix normal or canonical form (of a matrix)      140 164 369 372 374 399 437 438 559 723
Matrix notations      699
Matrix operations and their properties      702 703
Matrix operations and their properties: addition (sum)      702
Matrix operations and their properties: conjugate      702
Matrix operations and their properties: conjugate transpose      702
Matrix operations and their properties: multiplication      705
Matrix operations and their properties: power      707
Matrix operations and their properties: scalar multiplication      702
Matrix rank      89 125 366 383 407 711 722 734 739
Matrix rank, algebraic rank      125 368 383 407 739
Matrix rank, geometric rank      125 367 383 407 738
Matrix rank, geometric rank, column rank      125 367 383 407 711 738 757
Matrix rank, geometric rank, row rank      125 367 383 407 711 738 757
Matrix rank,equalities of ranks      125 367 383 407 711 739
Matrix representation (of linear transformation)      57 58 59 86 116 366 399 406 735
Matrix spectral radius      496
Matrix square root (of a matrix)      522 543
Matrix tensor product (of two matrices)      148
Matrix trace      107 122 713 714
Matrix transpose      700
Matrix, block or partitioned matrix      180; see “the index”
Matrix, column matrix (vector)      700 703
Matrix, diagonal matrix $\textrm{diag}[\mathrm{a}_1, \ldots , \mathrm{a}_\mathrm{n}]$       701
Matrix, equal      700
Matrix, generalized or pseudo inverse      see “the index”
Matrix, identity matrix $\mathrm{I}_\mathrm{n}$ (of order n)      701
Matrix, inner product for $\mathrm{m} \times \mathrm{n}$ real or complex matrices      714
Matrix, invertible or nonsingular (square) , inverse matrix      47 55 336 338 339 711
Matrix, invertible or nonsingular (square) matrix      47 55 336 338 339 711
Matrix, invertible or nonsingular (square) matrix, computation of inverse matrix      519
Matrix, invertible or nonsingular (square) matrix, computation of inverse matrix using Cayley — Hamilton theorem (eigenvalues)      108 399 793
Matrix, invertible or nonsingular (square) matrix, computation of inverse matrix using determinant and adjoint matrix      55 336 731
Matrix, invertible or nonsingular (square) matrix, computation of inverse matrix using elementary row operations      161 442 726
Matrix, invertible or nonsingular (square) matrix, expression as a product of finitely many elementary matrices      161 442 726
Matrix, lower triangular matrix      161 701
Matrix, matrix cosine      499
Matrix, matrix equation      712
Matrix, matrix exponential      496 551 558
Matrix, matrix logarithm      500
Matrix, matrix mth root      552
Matrix, matrix polynomial      551
Matrix, matrix power      550 552
Matrix, matrix sine      499
Matrix, power series of a matrix      499
Matrix, row matrix (vector)      700 703
Matrix, scalar matrix $\alpha \mathrm{I}_\mathrm{n}$       701
Matrix, similar      117 132 716
Matrix, similar equivalent relation      716
Matrix, similar, criteria of similarity of real matrices of order      3 517
Matrix, similar, necessary and sufficient conditions for similarity      198 808
Matrix, similar, necessary condition for a matrix similar to a diagonal matrix      198 718
Matrix, singular (square) matrix      712
Matrix, some peculiar matrices, adjoint matrix      336 731
Matrix, some peculiar matrices, companion matrix      572 574 796
Matrix, some peculiar matrices, elementary matrix      see “the index)”
Matrix, some peculiar matrices, Hermitian      701 782 786
Matrix, some peculiar matrices, idempotent      111 129 405 707
Matrix, some peculiar matrices, involutory      112 129 405 544 707
Matrix, some peculiar matrices, nilpotent      110 215 404 522 537 539 708 803
Matrix, some peculiar matrices, nilpotent, degree or index or power      215 537 539
Matrix, some peculiar matrices, normal      782 785
Matrix, some peculiar matrices, orthogonal      712 782 783 786
Matrix, some peculiar matrices, permutation matrix      160 767
Matrix, some peculiar matrices, skew-Hermitian      701 782 786
Matrix, some peculiar matrices, skew-symmetric      497 701 710 782
Matrix, some peculiar matrices, symmetric      700 710 782 786
Matrix, some peculiar matrices, unitary      782 783 786
Matrix, square matrix      700
Matrix, square matrix, (main) diagonal entries      700
Matrix, triangularizable      547
Matrix, triangularizable, decomposition as the sum of a diagonalizable matrix and a nilpotent matrix      549 550
Matrix, upper triangular matrix      161 701
Matrix, zero matrix      700
Menelaus theorem      291 292 648
Minimal polynomial      199 209 214 224 479 516 794 795
Multiplicity      199 797
Multiplicity, algebraic      199 480 516 797
Multiplicity, geometric      199 480 516 797
n-dimensional vector space      698
Natural coordinate system      see “Coordinate system”
Natural isomorphism      752
Natural or standard basis      see “Basis”
Natural or standard inner product      see “Inner product”
Natural projection      211 747
Natural projective basis      659
Natural projective basis, projective coordinate      659
Natural projective basis, unit      659
Natural projective basis, vertex      659
Negative definite matrix      789
Negative semidefinite matrix      789
Negatively-oriented or left-handed on $\mathbb{R}^2$       246
Newton identity      494
Nilpotent operator or matrix      see “Linear operator and matrix”
Noncentral conies      306
Noncentral quadrics      677
Nondegenerated conies      306
Nondegenerated quadratic surface      676
Nondegenerated quadrics      676
Nonhomogeneous linear equation      see “Inhomogeneous linear equation”
Nonsingular matrix      see “Matrix”
Norm or length of a vector      see “Inner product”
Normal form of a matrix      140 164 385 437 438 457 459 464 723
Normal operator or matrix      782 785;
Normal operator or matrix, Hermitian, symmetric      782 786
Normal operator or matrix, Hermitian, symmetric, indefinite      789
Normal operator or matrix, Hermitian, symmetric, positive, negative definite      786 789
Normal operator or matrix, Hermitian, symmetric, positive, negative semidefinite      786 789
Normal operator or matrix, orthogonally similar      717 784
Normal operator or matrix, skew-Hermitian, skew-symmetric      782
Normal operator or matrix, spectral decomposition theorem      786
Normal operator or matrix, unitarity similar      785 786
Normalized barycentric coordinate      76; see also “Barycentric coordinate”
Null space      89 366 734;
Nullity      89 366 734
Oblique (or affine) coordinate system      38 328
One-dimensional affine space      12
One-dimensional vector space      12
One-dimensional vector space, standard      12
One-way stretch      255 385 597
Orientation on $\mathbb{R}^1$       6
Orientation on $\mathbb{R}^2$       22 90 246
Orientation on $\mathbb{R}^2$ , anti (or counter) clockwise      22 90 246
Orientation on $\mathbb{R}^2$ , clockwise      22 90 246
Orientation on $\mathbb{R}^2$ , negatively-oriented or left-handed      246
Orientation on $\mathbb{R}^2$ , opposite orientation      246
Orientation on $\mathbb{R}^2$ , oriented      246
Orientation on $\mathbb{R}^2$ , positively-oriented or right-handed      246

Orientation on $\mathbb{R}^2$ , same orientation      246
Orientation on $\mathbb{R}^3$       320
Orientation on $\mathbb{R}^3$ , left-handed or clockwise      320
Orientation on $\mathbb{R}^3$ , right-handed or anticlockwise      320
Orthogonal complement (of a set in an inner product space)      129 412 778
Orthogonal decomposition theorem (of an inner product space)      778
Orthogonal matrix      266 272 313 461 613 625 783
Orthogonal matrix of order 2      169 266 272 313
Orthogonal matrix of order 3      461 466 613 625
Orthogonal matrix of order n      783
Orthogonal matrix of order n, characteristic properties      783
Orthogonal matrix of order n, orthogonally similar      786
Orthogonal matrix of order n, rigid motion or isometry      784
Orthogonal operator      782 783
Orthogonal operator, group of      783
Orthogonal or perpendicular (of two vectors)      129 169 774
Orthogonal projection      176 418 420 428 429 460 462 468 469 624 764 779
Orthogonal projection, characterizations      779
Orthogonal reflection      274 460 461 468 469 592 625
Orthogonal set or system      775 778
Orthogonal stretch      261 597
Orthonormal basis      176 461 613 621 625 775 776
parabola      303 304 305
Parabolic translation      314
Parabolic translation, transitive one-parameter subgroup of $\mathrm{G}_a (2;\mathbb{R})$       314
Paraboloid      670
Paraboloid, elliptic paraboloid      670
Paraboloid, hyperbolic paraboloid      671
Paraboloid, parabolic cylinder      671
Parallel law or invariance (of vectors)      7 25
Parallel lines in affine plane or space      63 303 304 349 645
Parallel lines in affine plane or space, two imaginary parallel lines      303 304 305
Parallel planes in affine space      355 645 671
Parallel planes in affine space, two imaginary parallel planes      671
Parallelogram, 2-parallelogram, i.e. a parallelogram usually called in $\mathbb{R}^2$       68
Parallelogram, 3-parallelogram, i.e. a parallelepiped usually called in $\mathbb{R}^3$       320 375 605 638 667
Parallelogram, k-parallelogram or k-hyperparallelepiped in $\mathbb{R}^\mathrm{n}$ ( $0 \leq \mathrm{k} \leq \mathrm{n}$ )      638 667
Parallelogram, k-parallelogram or k-hyperparallelepiped in $\mathbb{R}^\mathrm{n}$ ( $0 \leq \mathrm{k} \leq \mathrm{n}$ ), degenerated      638
Parallelogram, k-parallelogram or k-hyperparallelepiped in $\mathbb{R}^\mathrm{n}$ ( $0 \leq \mathrm{k} \leq \mathrm{n}$ ), side vectors      638 667
Parallelotope      661
Particular solution      see “Inhomogeneous linear equation”
Pasch axiom      295
Permutation matrix      160 767;
Perpendicular or orthogonal of two vectors      see “Orthogonal or perpendicular”
Pivot      162 721
Pivot matrix      162
Plane      351
Plane in affine or vector space $\mathbb{R}^3$       351
Plane in affine or vector space $\mathbb{R}^3$ coordinate equation w. r. t. a basis      352
Plane in affine or vector space $\mathbb{R}^3$ coordinate equations of coordinate planes in an affine base (coordinate system)      352 363
Plane in affine or vector space $\mathbb{R}^3$ directions      351 352
Plane in affine or vector space $\mathbb{R}^3$ parametric equation in vector form      352
Plane in affine or vector space $\mathbb{R}^3$ parametric equation w. r. t. a basis      352
Plane in affine or vector space $\mathbb{R}^3$ passing point or points      351 352
Plane in affine or vector space $\mathbb{R}^3$ , relative positions of a line and a plane (see line)      354
Plane in affine or vector space $\mathbb{R}^3$ , relative positions of two coincident      355 645
Plane in affine or vector space $\mathbb{R}^3$ , relative positions of two intersecting      355 645
Plane in affine or vector space $\mathbb{R}^3$ , relative positions of two parallel      355 645
Plane in affine or vector space $\mathbb{R}^3$ , relative positions of two planes      355 645
Polar decomposition (of a matrix)      772
Polyhedron      654
Polynomial      687
Polynomial degree      687
Polynomial division algorithm      688
Polynomial unique factorization theorem      689
Polynomial, equal      687
Polynomial, irreducible      688
Polynomial, polynomial function      687
Polynomial, relatively prime      688
Polynomial, vector space of polynomials of degree not greater than n      688 698
Polynomial, vector space of polynomials over a field      688 693
Positive definite matrix      786 789
Positive matrix      491 502 504
Positive semidefinite matrix      786 789
Positively-oriented or right-handed on $\mathbb{R}^2$       246
Projection (operator)      136 193 200 435 480 746
Projectionalization of a linear operator      137 436
Projective line $\mathrm{P}^1 (\mathbb{R})$       661
Projective plane $\mathrm{P}^2 (\mathbb{R})$       315 661
Projective plane $\mathrm{P}^2 (\mathbb{R})$ , infinite line      315
Projective plane $\mathrm{P}^2 (\mathbb{R})$ , infinite point or point at infinity or ideal point      315
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$       657 658
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , (k-1)-dimensional subspace at infinity $1 \leq \mathrm{k} \leq \mathrm{n}$       657
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , affine or inhomogeneous coordinate (of an affine point)      658
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , homogeneous coordinate (w. r. t. a basis)      658
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , hyperplane at infinity      657
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , natural projective basis      659
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , natural projective basis, projective coordinate      659
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , natural projective basis, unit      659
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , natural projective basis, vertex      659
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , point at infinity or ideal point      657 658
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , projective transformation      659
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , projective transformation, group of      660
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , projective transformation, matrix representation      659
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , representative vector      658
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , representative vector, ideal or infinite point      658
Projective space $\mathrm{P}^3 (\mathbb{R})$ or $\mathrm{P}^\mathrm{n} (\mathbb{R})$ , representative vector, ordinary or affine point      658
Pythagorean theorem (or formula)      22 778
Quadratic curve      300
Quadratic curve, center      306
Quadratic curve, classification in affine geometry      309
Quadratic curve, classification in affine geometry, affine invariants      310
Quadratic curve, classification in affine geometry, affinely equivalent      307
Quadratic curve, classification in affine geometry, canonical or standard forms (types)      303
Quadratic curve, classification in Euclidean geometry      303 305
Quadratic curve, classification in projective geometry      317
Quadratic curve, conics      300 306
Quadratic curve, conics, central, noncentral      306
Quadratic curve, conics, degenerated, nondegenerated      306
Quadratic curve, conics, reducible, irreducible      306
Quadratic form      304 455
Quadratic form as sum of complete square terms      456
Quadric or quadric surface in $\mathbb{R}^3$       456 668
Quadric or quadric surface in $\mathbb{R}^3$ center      676
Quadric or quadric surface in $\mathbb{R}^3$ center, central quadric      677
Quadric or quadric surface in $\mathbb{R}^3$ center, noncentral quadric      677
Quadric or quadric surface in $\mathbb{R}^3$ , canonical or standard forms      670
Quadric or quadric surface in $\mathbb{R}^3$ , canonical or standard forms in affine geometry (space $\mathbb{R}^3$ )      675
Quadric or quadric surface in $\mathbb{R}^3$ , canonical or standard forms in affine geometry (space $\mathbb{R}^3$ ), affine invariants      679
Quadric or quadric surface in $\mathbb{R}^3$ , canonical or standard forms in Euclidean geometry (space $\mathbb{R}^3$ )      671
Quadric or quadric surface in $\mathbb{R}^3$ , canonical or standard forms in projective geometry (space $\mathrm{P}^3 (\mathbb{R})$ )      679
Quadric or quadric surface in $\mathbb{R}^3$ , degenerated      676
Quadric or quadric surface in $\mathbb{R}^3$ , nondegenerated or proper      676
Quadric or quadric surface in $\mathbb{R}^3$ , regular point, singular point, double point      677
Quadric or quadric surface in $\mathbb{R}^3$ , regular surface      677
Quotient space      200 211 250 372 695
Range (subspace, space)      56 85 734
Rank      89 124 366 368 383 406 711 722 734 739
Rank decompoisition theorem of a matrix      184
Rank theorem for matrices or linear operators      140 164 436 438;
Rational canonical form      233 399 562 569 574 809;
Rational canonical form A-cycle (of a matrix A)      570 802
Rational canonical form annihilator      214 570 802
Rational canonical form companion matrix      571 574 796 802
Rational canonical form, rational canonical basis      233 562 574 813
Real field      12 685
Real general linear group on $\mathbb{R}^\mathrm{n}$       see “Group”
Real inner product space      773; see “Inner product”
Real vector space      29 692
Rectangular coordinate system      38 328;
Reflection      253 274 387 461 468 544 592; transformation”
Representative vector      658
Riesz representation theorem      780; see “Inner product space”
Right inverse      142 175 423 457 760
Right invertible      142 423 457 760
Right kernel      124 407 757
Rotation      272; see “Orthogonal transformation”
Row rank      125 367 383 407 711 739
Row space      125 407 757
Row vector      125 367 375 703

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